Differential Equations
Class 12 Mathematics · Mathematics Part II · Chapter 3
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Summary
A differential equation relates a function to its derivatives. Its order is the order of the highest derivative present, and its degree (when defined) is the power of that highest-order derivative once the equation is a polynomial in derivatives. A solution containing as many arbitrary constants as the order is the general solution; fixing the constants from initial conditions gives a particular solution. This chapter covers the formation of differential equations and three methods of solving first-order, first-degree equations: variables separable, where \(\dfrac{dy}{dx}=g(x)h(y)\) is rearranged and integrated; homogeneous equations, solved by the substitution \(y=vx\); and linear equations \(\dfrac{dy}{dx}+Py=Q\), solved using the integrating factor \(e^{\int P\,dx}\). Differential equations model growth, decay and many physical processes.
Order and degreeFormation of differential equationsVariables separable methodHomogeneous differential equationsLinear differential equations
Key terms
Order
The order of the highest derivative in the equation.
Degree
The power of the highest-order derivative when the equation is polynomial in derivatives.
General solution
A solution with as many arbitrary constants as the order.
Variables separable
Form \(\dfrac{dy}{dx}=g(x)h(y)\), solved by separating and integrating.
Homogeneous equation
Solved by the substitution \(y=vx\).
Integrating factor
\(e^{\int P\,dx}\) for the linear equation \(\dfrac{dy}{dx}+Py=Q\).
Extra questions & answers
Try one from this chapter
The order of a differential equation is: